Proofs by Contrapositiono A proof by contraposition is based on the logical equivalence between

a statement and its contrapositive. – Therefore, the implication p→ q can be proved by showing that its

contrapositive ~ q → ~ p is true. The contrapositive is usually proved directly.

o The method of proof by contrapositive may be summarized as:1. Express the statement in the form if p then q.

2. Rewrite this statement in the contrapositive form

if not q then not p.

3. Prove the contrapositive by a direct proof.

Proofs by Contrapositiono Prove that for all integers n, if n2 is even then n is even.

PROOF: The contrapositive of the given statement is:

“if n is not even (odd) then n2 is not even (odd)”

We prove this contrapositive statement directly.

Suppose n is odd. Then n = 2k + 1 for some k Z

Now n2 = (2k+1) 2= 4k2 + 4k + 1

= 2•(2k2 + 2k) + 1

= 2•r + 1 where r = 2k2 + 2k Z

Hence n2 is odd. Thus the contrapositive statement is true and so the given statement is true.

Proofs by ContrapositionFor all integers m and n, if m + n is even then m and n are both even or m and n are both odd.PROOF:The contrapositive statement is:“For all integers m and n, if one of m and n is even and the other is odd, then m + n is odd”Suppose m is even and n is odd. Then

m = 2p for some integer p

and n = 2q + 1for some integer qNow m + n = (2p) + (2q + 1)

= 2•(p+q) + 1

= 2•r + 1 where r = p+q is an integer

Hence m + n is odd.Similarly, taking m as odd and n even, we again arrive at the result that m + n is odd.Thus, the contrapositive statement is true. Since an implication is logically equivalent to its contrapositive so the given implication is true.

Proofs by ContrapositionShow that if 3n + 2 is an odd integer,then n is odd. Proof : Assume that n is even. This implies that n = 2k for some integer k. Then, 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k + 1), so that 3n + 2 is even. Since the negation of conclusion implies the negation of hypothesis, the original conditional statement is true.

Homework1. Prove that if n2 is not divisible by 25, then n is not divisible by 5.

2. Prove that if n is an integer and n3 + 5 is odd, then n is even.

3. Prove that if 3n + 2 is odd, then n is odd